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It's not intuitively clear to me how the game can go on forever -- I would expect that, eventually, you would hit upon some valid pattern. The explanation in the text didn't really make sense to me. Could anyone help with this?

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The game could use a better definition of what constitutes a pattern.

>> A pattern is a sequence of pebbles repeated three times in a row.

By that logic, player 2 would have lost at their fifth turn. ... And 'in a row' is open to interpretation since it is additional information to 'repeated three times'.

I think it's pretty clear that a string s of pebbles is a pattern iff there exist strings x and y such that s = x|y|y|y, where | denotes string concatenation and y is nonempty.

EDIT: Clarify that only y is nonempty.

I was confused also. My guess was that maybe a “sequence” has to be 4 or more pebbles?

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